Special Relativity

What does ‘relativity’ mean? According to a common dictionary, it means that something is dependent on or relative to something else.  So, natural questions arise: when and to whom is it relative? What is relative? Why is it relative?

(If you prefer you can read this article in Portuguese: Relatividade Restrita.)

•  When and to whom is it relative?

The “special relativity” is applicable to two observers with different movements. By “observers” I mean something which can measure physical quantities (e.g., position, velocity, etc.).

Let us suppose that we are driving a car on the highway. We are traveling at 120 km/h and some irresponsible (or criminal) passes by us at 200 km/h. It’s clear that he is traveling 80 km/h faster than us. This means that his speed is 80 km/h relatively to us.

We observe the world from our referential, so, obviously, we are always at rest relatively to ourselves. If we are traveling by car and we pass by a parked car, we actually see the parked car passing by us. From a physical point of view, both observations are correct, it’s just a matter of different perspectives (referential). Notice that we are never at “absolute rest”, because our planet moves, the Solar System moves, the galaxy is also moving… So, why should the Earth be a special referential? It is not special. we just commonly use it because the Earth looks at rest from our common sense, however, it’s equally correct to consider any other reference frame when we wish to measure a movement. If each individual would consider himself as its own referential, then it is clear that different people would describe the world differently. One could say that his car was at rest, while another person could say it was moving, and both would be correct. In the previous example, I can say that the imbecile is moving at a speed of 80 km/h (relatively to me), and someone at rest near the highway will say that he is moving at 200 km/h. We are both right. All of this to say that there are no privileged reference frames. Furthermore, the laws of Physics should hold for everyone, independently on their movements.

• What is relative and why is it?

Taking into account what I just said, speed is relative. Note, however, that speed can be described as a distance traveled during a certain time. So, if speed is relative, does it imply that distances or times are relative? If we consider that time is absolute (i.e. time does not depend on the referential that we choose to measure it), like Galileo Galilei assumed, then space has to be relative! For instance, I am always at rest relative to myself, therefore my speed is zero, and my “traveled distance” is also zero. However, another observer will disagree, because he had seen me moving. This doesn’t imply anything really astonishing, for it’s just a matter of doing a simple transformation from one reference frame to another. Obviously, the spatial distances are invariant from my perspective (which is perfectly fine if we the speed is sufficiently low, as we will see). One can say that these transformations between reference frames are just simple mathematical “tricks”, which don’t give us any new insight about the underlying Physics, or anything out of the ordinary.

Fortunately for us, the Theory of Relativity tells us that this is not as simple as this. Maxwell Laws of electromagnetism do not follow these simple rules that I just presented, for they predict that light travels at a constant speed in any reference frame that we may choose! It doesn’t matter if you are at rest, or walking: when you measure the speed of light, you will always measure approximately 300 000 km/s, c (this is the speed of light in the vacuum, for in different media light travels at lower speeds). Following common sense, as illustrated above, you would expect to measure a lower speed if you would be chasing the light: let’s say that you are running at a speed of ‘x’ and you measure the speed of light. You would expect to measure ‘c-x’. However, you measure ‘c’, independently on your speed! All experimental evidence has confirmed this weird fact.

So, we have already found the two fundamental postulates of Einstein’s Special Relativity proposed in 1905:

First, the Laws of Physics do not depend on the chosen referential. For this theory to apply, there is, however, one condition: the reference should be an “inertial” reference, so-called “inertial frame”, which means that the referential has to have a constant speed (the constant can be zero). Acceleration demands another theory, the General Relativity. The second postulate is that the speed of light in the vacuum is constant and independent on the referential.

These two postulates imply that not only space is relative, but also time is relative! 

Why? Let us consider the following example:

Take an idealized system of two parallel mirrors separated by a distance ‘d’. Then, consider a light beam (which we can measure) moving perpendicularly between the mirrors (without being able to escape, or being absorbed by the mirrors). Then, we put this system moving at a high speed (the direction of this movement is parallel to the mirrors positions, i.e, perpendicular to the light beam). In this scenario, an observer at rest should see the light not moving in a perpendicular “line” relative to the mirrors, but following an oblique direction (otherwise, the light beam would escape from the mirrors, given that when the light departs from one mirror, it has to target the position that the other mirror will have after the time taken to move from one mirror to the other). Another observer, who follows the system of mirrors, will see the light moving perpendicularly relative to the mirrors, given that in this case, the system is at rest relative to him. Note that if light follows the perpendicular to the mirrors, the speed of light c is equal to the distance ‘d’ divided by the time ‘t1’ spent by the beam to travel that distance. So c=d/t1. In the other case, when the light has to travel an oblique path, the speed is the ratio of ‘x’, the diagonal observed, and the time ‘t2’ spent by the beam to travel ‘x’. So c=x/t2. At this point, you may incorrectly assume that t1 is equal to t2.

Note that by it is clear that ‘x’ is larger than ‘d’ (the diagonal is larger than the perpendicular), therefore, since ‘c’ is constant, it means that ‘t1’ has to be different (smaller) than ‘t2’!

Both observers measure the time in the same way, the time interval between two events: the light beam departs from one mirror, and the light beam reaches the other mirror. If they measure different times, this means time depends on the chosen inertial frame.

Usually, people think that the observer at rest is being fooled by some kind of optical illusion, i.e. they think that there is no diagonal. That’s not true, he is as correct as the other observer (they can use the same device to measure times and distances). They are both using the same Laws of Physics.

One striking consequence of this is that what one observer considers as simultaneous events may not be simultaneous for another observer. This sounds very strange and completely against our common sense for we never observe such kind of phenomena in our day-to-day life. The reason why we don’t observe this is because relativity only becomes noticeable at speeds close to the speed of light (how close? That depends on how accurate you want to be with your measurements). Our day-to-day life is just too slow for us to realize such phenomena. Einstein proposed an “experiment” which we could do if that wouldn’t be the case, in other words, an experiment for us to imagine:

Let us consider a light source (the Sun, for instance) and a train traveling very fast. Suppose that an observer at rest (outside of the train) can see two light beams reaching the train: one at the front and another at the back of the train (as in the figure above). If these two light beams came at the same time from the Sun, then this observer will see them both reaching the train at the same time.

Then, suppose that there is another observer, this one inside the train (say, in the middle of the train). He will see first a beam reaching the front of the train, for he is moving towards the beam, together with the train, and only later he will see the other beam reaching the back. Thus, the two observers won’t agree concerning the simultaneity of the two events.

It may be convenient to explain in more detail this “imaginary experiment” (if you understood it, you can skip this paragraph). First, an observer sees the light that reaches his eyes; second, the light that arrived at the front of the train is reflected in all directions (at the back of the train the same happens). As a consequence, a moving observer can get the light coming from these two sources at different instants: if he is approaching one of the sources while moving away from the other, then he should see first the light coming from the one to which he is moving towards to.

(Now I will present some simple mathematical formulas about this. If you are not interested in this more technical aspects, you can continue to read below*.)

There are simple equations that can be obtained by “playing” with the previous postulates. One of the most significant describes the time distortion:

where

‘t’ is the time measured by an observer at rest, ‘to’ is the time measured by an observer with speed ‘v’, and ‘c’ is the speed of light.

Similarly, it is possible to derive the equation for the spatial distortion (contraction):

where ‘L’ is the measured distance by an observer at rest and ‘Lo’ is the distance measured by an observer with speed ‘v’.

By studying these equations and taking into account that the speed ‘v’ is always smaller than the speed of light ‘c’ (which can be considered as a maximum speed imposed by nature), it is easy to understand that gamma will be always larger than one (approximately one when special relativity is not relevant, i.e., when ‘v’ is small compared to ‘c’).

*The formulas above show that the time measured by an observer at rest is larger than the time measured by a traveler, which means that a clock of a traveling observer gets late in comparison to a clock at rest. This fact has been observed experimentally. It is known as the “twin paradox”: suppose that two twins are born in our planet Earth, then they are separated, one stays with us (having a “normal” life), while the other twin travels in a spacecraft at high speed (say, for instance, at 90% of the speed of light). When they meet again, their age is no longer the same! The traveler is younger! (You can calculate how much younger using the formula above, and assuming that he traveled for, say, 10 years.)

By analyzing the equation for spatial distortion, we find spatial contraction, for gamma is larger than one, making ‘L’ smaller than ‘Lo’. Thus, the observer at rest sees objects contracting when they are traveling at high speed, whilst the moving observer will see the same spatial contraction in objects at rest, given that for him, he is not moving, everything else is.)

However, if we consider low speeds, gamma is in a good approximation one, and therefore distances and times are almost invariant in different reference frames, in agreement with our common sense.

Thus, Special Relativity tells us that both space and time are relative, which may become obvious depending on the speed.

 

Please use the comments section below to ask me questions, make remarks, etc.
Note that English is not my native language, so please feel free to propose corrections or improvements to the text. Thank you.

Marinho Lopes